Problem 37 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$8 x^{2}-2 x-1$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(8x^{2}-2x-1\) is \((4x+1)(2x-1)\).

Step 1: Identifying Suitable Factor Pairs

For the given trinomial \(8x^{2}-2x-1\), find two numbers that added together equal -2 (the coefficient of x), and when multiplied, result in -8 (the product of the coefficients of the \(x^{2}\) term and the constant term). These numbers are -4 and 2.

Step 2: Factoring the Trinomial

Substitute x-coefficient (-2x) with -4x + 2x in the trinomial and factor by grouping. So, \(8x^{2}-2x-1\) becomes \(8x^{2}-4x+2x-1\) which equals \(4x(2x-1) + (2x-1)\). Since \(2x-1\) is common, factor it out to get \( (4x+1)(2x-1)\).

Step 3: Checking the Factorization with FOIL

Multiply the factored trinomial using FOIL (First, Outside, Inside, Last): First terms are \(4x * 2x = 8x^{2}\), Outside terms are \(4x * -1 = -4x\), Inside terms are \(1 * 2x = 2x\) and Last terms are \(1 * -1 = -1\). Add these up to yield \(8x^{2} - 2x - 1\) which matches the original trinomial, thus confirming the factorization is correct.

Key Concepts

Polynomials

When we talk about polynomials, we're referring to expressions that can have multiple terms, usually involving variables and coefficients. In particular, a trinomial is a type of polynomial that has exactly three terms. For example, in the original exercise, we had the trinomial \(8x^2 - 2x - 1\).

These terms are defined by specific parts:

• The "\(8x^2\)" term is known as the quadratic term because it involves the square of the variable.
• The "\(-2x\)" is the linear term since it has the variable to the first power.
• Lastly, "\(-1\)" stands as the constant term since it doesn't involve the variable.

Understanding polynomials helps you recognize how they can be manipulated, specifically how they can be factored, which is a key algebraic skill. Polynomials can be factored into products of simpler polynomials or numbers, making them easier to work with.

FOIL Method

The FOIL method is a valuable technique used to multiply two binomials. This method is straightforward and helps in verifying the factorization of polynomials. FOIL stands for First, Outside, Inside, and Last, referring to the pairs of terms you multiply together:

• First: Multiply the first terms of each binomial. For instance, in \((4x+1)(2x-1)\), multiply \(4x\) and \(2x\) to get \(8x^2\).
• Outside: Multiply the outer terms. Here, \(4x\) and \(-1\) combine to give \(-4x\).
• Inside: Multiply the inner terms, such as \(1\) and \(2x\), resulting in \(2x\).
• Last: Multiply the last terms of the binomials. For these, \(1\) and \(-1\) produce \(-1\).

After calculating these products, you sum them: \(8x^2 - 4x + 2x - 1\). Simplifying gives \(8x^2 - 2x - 1\), confirming the factorization is correct and has brought us back to the original trinomial.

Factoring by Grouping

Factoring by grouping is a strategy for simplifying high-order polynomials by breaking them down into smaller groups. This method is particularly useful when working with trinomials that don’t factor easily into simple binomials.

With the trinomial \(8x^2 - 2x - 1\), we started by identifying and substituting the \(-2x\) term with \(-4x + 2x\). This way, we split the middle term to allow for easier grouping. The expression now reads: \(8x^2 - 4x + 2x - 1\).

Next, you group terms to factor common elements:

• From the first group \(8x^2 - 4x\), factor out \(4x\) to get \(4x(2x-1)\).
• From the second group \(2x - 1\), notice it remains as it cannot be simplified further, but matches the factor \((2x-1)\) from the first group.

Once both parts have a common binomial \((2x-1)\), you can factor that out, leading to the final result: \((4x+1)(2x-1)\).

This method efficiently simplifies complex expressions into easily manageable ones, facilitating both the solving of equations and verification by the FOIL method.